Zonotopes

Zonotopes are a sub-class of polytopes defined as the image of a unit cube under an affine transformation. For example,

B = BallInf(zeros(2), 1.0) # unit cube
M = [0 1; -1 0] # an affine transformation

Z = linear_map(M, B) # zonotope
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([0.0, 0.0], [0.0 1.0; -1.0 0.0])

Zonotopes are commonly represented using their generator representation. Here, a zonotope $Z ⊆ \mathbb{R}^n$ is defined by a center $c ∈ \mathbb{R}^n$ and a finite number of generators $g_1, . . . , g_p ∈ \mathbb{R}^n$ such that

$$$Z = \left\{ c + \sum_{i=1}^{p} ξ_i g_i | ξ_i ∈ [−1, 1]\right\}.$$$

It is common to note $Z = (c, ⟨ g_1 . . . , g_p ⟩)$ or simply $Z = (c, G)$, where $g_i$ is the $i$-th column of $G$. In the example above,

@show center(Z)

@show genmat(Z)
2×2 Matrix{Float64}:
0.0  1.0
-1.0  0.0

The generators can be

Z3 = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 0.0], [0.1 0.0; 0.0 0.1])

The order of a zonotope is the ratio between the number of dimensions and the number of generators; i.e. $o = \frac{k}{n}$. Use the function order(Z) to get the order of a given zonotope.

Z = Zonotope([1, 1.], [-1 0.3 1.5 0.3; 0 0.1 -0.3 0.3])
plot(Z)
quiver!(fill(1., 4), fill(1., 4), quiver=(genmat(Z)[1, :], genmat(Z)[2, :]), color=:black)

Other characterizations

There are other useful characterization of zonotopes. A zonotope can be seen as the Minkowski addition of line segments resulting in centrally symmetric convex polytopes as shown in the following figure, which illustrates how each generator spans the zonotope.

Operations with zonotopes

The cost is measured in terms of the number of binary operations, $\mathrm{Op}(⋅)$.

$Z_1 = (c, ⟨ v_1, …, v_k ⟩), Z_2 = (d, ⟨ w_1, …, w_m ⟩) ⊆ \mathbb{R}^n, M ∈ \mathbb{R}^{m \times n}$

$Z_1 \oplus Z_2 = (c+d, ⟨ v_1, …, v_k, w_1, …, w_m ⟩)$

$MZ_1 = (Mc, ⟨ Mv_1, …, Mv_k ⟩)$

$CH(Z_1, e^{Aδ}Z_1) ⊆ \frac{1}{2}(c + e^{Aδ}c,⟨ v_1 + e^{Aδ}v_1, …, v_k+e^{Aδ}v_k, v_1 - e^{Aδ}v_1, v_k - e^{Aδ}v_k, c - e^{Aδ}c ⟩ )$

OperationSimplification RuleCost
$Z_1 \oplus Z_2$$n MZ_1$$2mn(k+1)$
$CH(Z_1, e^{Aδ}Z_1)$$2n^2(k+1)+2n(k+2)$